Q: Which of the following limits does NOT exist?

$$\lim\limits_{x \to 0} \frac{1}{x}$$

Question 1

Given the function $$f(x) = 2x^2$ - 3x + 5$$, what is the vertex of the parabola?

Accepted Answer

$$(\frac{3}{4}, \frac{31}{8})$$

Question 2

A company’s cost function is $$C(x) = 500 + 20x$$, and its revenue function is $$R(x) = 50x. $$What is the profit when 30 units are sold?

Accepted Answer

$$P(30) = 900$$

Question 3

Which of the following functions is discontinuous at $$x = 2$$?

Accepted Answer

$$f(x) = \frac{x^2$ - 4}{x - 2}$$

Question 4

Find the slope of the line passing through the points $$(2, 5)$$ and $$(6, 13).$$

Accepted Answer

$$2$$

Question 5

Suppose $$f(x) is $$defined as follows:

$$\begin{cases} x^2$, & x \u003c 1 \ 2x + 1, & x \geq 1 \end{cases}$$

Does $$\lim\limits_{x 	o 1} f(x)$$ exist?

Accepted Answer

No, because the left and right limits are not equal.

Question 6

If $$f(x) = 3x^3$ - 2x + 7$$, what is $$f'(x)$$?

Accepted Answer

$$9x^2 - 2$

Question 7

A medication decays in the bloodstream according to $$A(t) = 100e$$^{-0.2t}$$. What is the instantaneous rate of change of the medication amount at t = 5$ hours?

Accepted Answer

$$A'(5) = -20e$$^{-1}$$

Question 8

Which of the following best describes the derivative of a position function $$s(t)$$ with respect to time?

Accepted Answer

It gives the velocity at time $$t.$$

Question 9

Evaluate $$\lim\limits_{x 	o \infty} \frac{5x^2$ + 3}{2x^2$ - x}.$$

Accepted Answer

$$\frac{5}{2}$$

Question 10

A savings account earns continuous interest at a rate of $$4\%$$ per year. If you deposit $$\2000$, how much will you have after 10$ years? (Use A$$ = $$Pe^{rt}$)

Accepted Answer

$$A = 2000e$$^{0.4}$$ \approx 2984.18$$

Question 11

Which of the following statements is true about the derivative at a sharp corner (cusp) on a graph?

Accepted Answer

The derivative does not exist at that point.

Question 12

Given $$f(x) = e$$^{2x}$$, what is f$$'(x)$$?

Accepted Answer

$$2e^{2x}$$

Question 13

A company’s marginal cost function is $$C'(x) = 5 + 0.1x. $$What does $$C'(50)$$ represent?

Accepted Answer

The approximate cost to produce the 51st unit.

Question 14

Find the equation of the line with slope $$-3$$ passing through the point $$(4, 2) in $$slope-intercept form.

Accepted Answer

$$y = -3x + 14$$

Question 15

If $$f(x) = x^2$ - 4x + 3$$, what are the $$x-$$intercepts?

Accepted Answer

$$x = 1$$ and $$x = 3$$

Question 16

Which of the following best describes the domain of $$f(x) = \frac{1}{x-5}$$?

Accepted Answer

All real numbers except $$x = 5$$

Question 17

Suppose $$f(x) = x^2$ and g(x) = 2x + 1$. What is (f$$ \circ g)(2)$$?

Accepted Answer

$$25$$

Question 18

If the revenue function is $$R(x) = 100x - x^2$, what is the marginal revenue when x = 10$?

Accepted Answer

$$80$$

Question 19

Which of the following limits does NOT exist?

Accepted Answer

$$\lim\limits_{x 	o 0} \frac{1}{x}$$

Question 20

A ball is thrown upward and its height (in meters) after $$t$$ seconds is $$h(t) = 20t - 5t^2$. What is the velocity after 2$ seconds?

Accepted Answer

$$0 m/s$$

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